onsider a market consisting of a single stock and a money market account
(MMA). The MMA is riskless and has rate
.
The price of the stock is given by a stochastic process
.
At present moment
the stock is priced at
.
There is no trading or price evolution until a time moment
,
and at
the stock may assume only two possible prices:
and
.
Hence, we introduce the random
events
We would like to come up with a
price
for a contract that pays at time
a random
amount
Suppose that at the moment
we sell the contract for a price
,
purchase
shares of the stock and put the remainder of the funds
into the MMA. Any of the values
may be negative. We assume that we can short the stock and borrow from MMA at
the same rate
.
The value of the position
is
zero:
We hold the position until the time moment
when there are two
possibilities:
Let us select
and
so that
would be the same for both
situations:
We arrive to a system
of two
equations:
The
quantity
is called "delta" and the strategy of taking position in the underlying stock
equal to
is called "delta hedging".
To determine
we make the following "no arbitrage" argument. At
we have a position of zero value. At the moment
we have a position of set value (given our selection of
).
Such set value has to be zero because otherwise we made or lost money in
absence of risk, hence, the price
of the contract would not be correct. Therefore, we have a third
equation:
from which we derive
:
and we obtain the correct price of the contract:
We would like to put the above result into the
form
for some numbers
.
Hence, we rearrange the expression
:
thus
Note
that
For this reason we call the numbers
the "risk neutral
probabilities":
and represent the expression
as the "risk neutral expectation of discounted
payoff":
